March 2012 Legg Mason Capital Management Shaking the Foundation

Perspectives March 2012 Legg Mason Capital Management Shaking the Foundation Past performance is no guarantee of future results. All investments involve risk, including possible loss of principal. This material is not for public distribution outside the United States of America. Please refer to the disclosure information on the final page. In the unIted states – InVestMent PROduCts: nOt FdIC InsuRed • nO BanK GuaRantee • MaY LOse VaLue Batterymarch I Brandywine Global I ClearBridge Advisors I Legg Mason Capital Management Legg Mason Global Asset Allocation I Legg Mason Global Equities Group I Permal Royce & Associates I Western Asset Management March 23, 2012 Shaking the Foundation Revisiting Basic Assumptions about Risk, Reward, and Optimal Portfolios An Interview with Ole Peters Investing is fundamentally a bet on the future. The problem is that we only get to live once: we have to make choices in the face of uncertainty and deal with the consequences for better or worse. The mean-variance approach, the most common guide to build a portfolio, basically says that you should get the most return for a given level of risk based on the expected values of the individual assets and on how they correlate with one another. One of the great aspects of an affiliation with the Santa Fe Institute (SFI) is the opportunity to exchange ideas with world class scientists who are self-selected to be curious about the world. Ole Peters, trained as a physicist and a visiting researcher at SFI, is a great example. Ole has been discussing his concerns about the standard theories in economics for years, including portfolio theory, and he gave a talk last fall that prodded me to share his thoughts with a wider audience. At first, I considered writing about this myself. But then I figured it would be better to interview Ole and let you hear the story directly from him. Fair warning: this is not easy material. But I believe working through these ideas and their implications is time well spent. Here’s a brief summary: • Distinction between ergodic and non-ergodic systems. Ole starts us off with this crucial distinction. An ergodic system is one where the ensemble average and the time average are the same. For example, the proportion of heads and tails is the same either if you ask 1,000 people to flip a coin at the same time (ensemble average because an ensemble of people are flipping simultaneously) or if you flip it yourself 1,000 times (time average because it takes time to flip sequentially). His point is that many of the models that economists use were designed to deal with ergodic systems, yet the reality is that we live in a world that is non-ergodic. That mismatch is problematic for portfolio construction. • How theory evolves is important. Over the last 150 years, economists have borrowed ideas from physics—generally with the goal of making economics into a harder science. In the late 1800s, Ludwig Boltzmann described ergodicity, but recognized that it worked under very narrow conditions. Economists embraced the “ergodic hypothesis” and applied it to systems that are not ergodic. The stock market is a good example. • Optimal leverage. In a mean-variance approach, risk and reward are related to one another in a linear fashion: more risk, more potential reward. And the main way to increase risk is to add leverage through debt. The key is that the ensemble average gets “rid of fluctuations before the fluctuations really have any effect.” If you assume a time average and a multiplicative process—that is, you parlay your bets—then there is an optimal amount of leverage. Too little leverage leaves potential profit on the table, but too much leverage assures ruin. Page 1 Legg Mason Capital Management MM: Ole, thanks for taking the time to share your thoughts with us. In many fields, there are foundational assumptions that, once established, are often not discussed or critically examined. Because these foundations are laid by the intellectual leaders in the field, they tend to go unquestioned. You have been doing some fascinating work that challenges some of the foundational assumptions in economics and finance. One such assumption is called the "ergodic hypothesis." Can you explain the difference between an ergodic and a non-ergodic system and tell us why this is so relevant in economics and finance? OP: Thank you, Michael. It's a pleasure to discuss this with you. Ergodicity is this complicated-sounding word, and a lot of highly technical work has been done in ergodic theory, but as usual the real meat is conceptual. The conceptual part is deep and subtle and utterly fascinating, but it's not actually very hard to understand. That doesn’t mean we can leave out the mathematics. If you want to use the concept, you have to understand the mathematics—there’s “no royal road”, as Luca Pacioli put it. Without the mathematics, you can’t get beyond what I call an “incomplete treacherous intuition” of the meaning. But since this is an interview, not a mathematics lecture, I will try to convey that incomplete treacherous intuition. Here it is: In an ergodic system time is irrelevant and has no direction. Nothing changes in any significant way; at most you will see some short-lived fluctuations. An ergodic system is indifferent to its initial conditions: if you re-start it, after a little while it always falls into the same equilibrium behavior. For example, say I gave 1,000 people one die each, had them roll their die once, added all the points rolled, and divided by 1,000. That would be a finite-sample average, approaching the ensemble average as I include more and more people. Now say I rolled a die 1,000 times in a row, added all the points rolled and divided by 1,000. That would be a finite-time average, approaching the time average as I keep rolling that die. One implication of ergodicity is that ensemble averages will be the same as time averages. In the first case, it is the size of the sample that eventually removes the randomness from the system. In the second case, it is the time that I’m devoting to rolling that removes randomness. But both methods give the same answer, within errors. In this sense, rolling dice is an ergodic system. I say “in this sense” because if we bet on the results of rolling a die, wealth does not follow an ergodic process under typical betting rules. If I go bankrupt, I’ll stay bankrupt. So the time average of my wealth will approach zero as time passes, even though the ensemble average of my wealth may increase. A precondition for ergodicity is stationarity, so there can be no growth in an ergodic system. Ergodic systems are zero-sum games: things slosh around from here to there and back, but nothing is ever added, invented, created or lost. No branching occurs in an ergodic system, no decision has any consequences because sooner or later we'll end up in the same situation again and can reconsider. The key is that most systems of interest to us, including finance, are non- ergodic. It's probably best to step back and start at the beginning. The word "ergodicity" was coined during the development of statistical mechanics. Ludwig Boltzmann, an Austrian physicist, invented it. He called ergodic systems "monodic" at first. Here’s the etymology: a "monodic" system is one that possesses only one (mon-) path (-odos) through the space of its possible states. The word became "ergodic" because Boltzmann considered systems of fixed energy (or work = ergon), and the idea was that the one path covers all the states allowed by that energy (the energy shell). Page 2 Legg Mason Capital Management If this is the case, Boltzmann argued that the system will over time—in his case a second may be long enough—visit all the relevant states that it can access. The system will visit each state with some relative frequency. We can mathematically treat those frequencies as probabilities, which has the incredibly nice consequence that the long-time averages of the quantities we're interested in are formally the same as expectation values arising from the relative frequencies that we interpreted as probabilities. Just like in the dice example. This makes the mathematics extremely simple. But it is a trick (Boltzmann literally called it a trick): we calculate expectation values that are a priori 1 irrelevant. In this very special case of an ergodic system, expectation values are the same as time averages, and that's why we're interested in these special cases. Time averages are interesting because they are what we observe in physics. We usually measure some sort of macroscopic property, like the pressure in a balloon. That pressure, in terms of the microscopic model, is the rate of momentum transfer per unit area to the balloon membrane resulting from a gazillion collisions of molecules with the membrane. Any device that we use to measure this is so sluggish that it will only give us a long-time average value of that momentum transfer. The very clever insight of Boltzmann was that under very special conditions, we just have to calculate an expectation value of the rate of momentum transfer per area, and that will coincide with the time average pressure that we actually observe. So, practically, ergodic means that time averages are the same as ensemble averages, or expectation values. Non-ergodicity means that they are different. Since there are many more ways of being different than there are of being identical, it comes as no surprise that most systems are non-ergodic.

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